# American Institute of Mathematical Sciences

November  2019, 13(4): 601-612. doi: 10.3934/amc.2019037

## A network reliability approach to the analysis of combinatorial repairable threshold schemes

 David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

* Corresponding author: Douglas R. Stinson

Received  November 2018 Revised  January 2019 Published  June 2019

Fund Project: The second author is supported by NSERC discovery grant RGPIN-03882.

A repairable threshold scheme (which we abbreviate to RTS) is a $(\tau,n)$-threshold scheme in which a subset of players can "repair" another player's share in the event that their share has been lost or corrupted. This will take place without the participation of the dealer who set up the scheme. The repairing protocol should not compromise the (unconditional) security of the threshold scheme. Combinatorial repairable threshold schemes (or combinatorial RTS) were recently introduced by Stinson and Wei [8]. In these schemes, "multiple shares" are distributed to each player, as defined by a suitable combinatorial design called the distribution design. In this paper, we study the reliability of these combinatorial repairable threshold schemes in a setting where players may not be available to take part in a repair of a given player's share. Using techniques from network reliability theory, we consider the probability of existence of an available repair set, as well as the expected number of available repair sets, for various types of distribution designs.

Citation: Bailey Kacsmar, Douglas R. Stinson. A network reliability approach to the analysis of combinatorial repairable threshold schemes. Advances in Mathematics of Communications, 2019, 13 (4) : 601-612. doi: 10.3934/amc.2019037
##### References:
 [1] J. Benaloh and J. Leichter, Generalized secret sharing and monotone functions, Lecture Notes in Computer Science, 403 (1990), 27-35 (CRYPTO '88 Proceedings). doi: 10.1007/0-387-34799-2_3.  Google Scholar [2] C. J. Colbourn, The Combinatorics of Network Reliability, Oxford University Press, 1987.   Google Scholar [3] C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007.  Google Scholar [4] B. Kacsmar, Designing Efficient Algorithms for Combinatorial Repairable Threshold Schemes, Masters Thesis, University of Waterloo, 2018. Google Scholar [5] T. M. Laing and D. R. Stinson, A survey and refinement of repairable threshold schemes, Journal of Mathematical Cryptology, 12 (2018), 57-81.  doi: 10.1515/jmc-2017-0058.  Google Scholar [6] A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar [7] D. R. Stinson and M. B. Paterson, Cryptography. Theory and Practice, Chapman & Hall/CRC, Boca Raton, 2019. Google Scholar [8] D. R. Stinson and R. Wei, Combinatorial repairability for threshold schemes, Designs, Codes and Cryptography, 86 (2018), 195-210.  doi: 10.1007/s10623-017-0336-6.  Google Scholar

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##### References:
 [1] J. Benaloh and J. Leichter, Generalized secret sharing and monotone functions, Lecture Notes in Computer Science, 403 (1990), 27-35 (CRYPTO '88 Proceedings). doi: 10.1007/0-387-34799-2_3.  Google Scholar [2] C. J. Colbourn, The Combinatorics of Network Reliability, Oxford University Press, 1987.   Google Scholar [3] C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007.  Google Scholar [4] B. Kacsmar, Designing Efficient Algorithms for Combinatorial Repairable Threshold Schemes, Masters Thesis, University of Waterloo, 2018. Google Scholar [5] T. M. Laing and D. R. Stinson, A survey and refinement of repairable threshold schemes, Journal of Mathematical Cryptology, 12 (2018), 57-81.  doi: 10.1515/jmc-2017-0058.  Google Scholar [6] A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar [7] D. R. Stinson and M. B. Paterson, Cryptography. Theory and Practice, Chapman & Hall/CRC, Boca Raton, 2019. Google Scholar [8] D. R. Stinson and R. Wei, Combinatorial repairability for threshold schemes, Designs, Codes and Cryptography, 86 (2018), 195-210.  doi: 10.1007/s10623-017-0336-6.  Google Scholar
Expected number of repair sets for ${\mathtt{SQS}}(v)$
 size type expected number 2 $3(r_2-1)^2 p^2$ 3 pair-pair-pair $4(r_2-1)^3 p^3$ 3 pair-pair-point $12(r_2-1)^2(r_1-3r_2+2) p^3$ 3 pair-point-point $6(r_2-1)(r_1-3r_2+2)^2 p^3$ 4 $(r_1-3r_2+2)^4 p^4$
 size type expected number 2 $3(r_2-1)^2 p^2$ 3 pair-pair-pair $4(r_2-1)^3 p^3$ 3 pair-pair-point $12(r_2-1)^2(r_1-3r_2+2) p^3$ 3 pair-point-point $6(r_2-1)(r_1-3r_2+2)^2 p^3$ 4 $(r_1-3r_2+2)^4 p^4$
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